Can I run Docker Desktop on Windows without admin privileges?

I have not tested it, but this article seems to indicate that you can.

TLDR;

Problem

The reason for requiring an admin session is that the Docker client in the default configuration uses a named pipe to connect to the Docker engine and that named pipe can only be accessed by administrators

Solution

To avoid this, you can simple allow your user FullControl access to that named pipe

Original author also provides a powershell tool with sources to help perform this admin task.

As of Docker Desktop 2.4 (possibly earlier, I haven't tested) this is supported. From the Windows installation instructions:

If your admin account is different to your user account, you must add the user to the docker-users group. Run Computer Management as an administrator and navigate to Local Users and Groups > Groups > docker-users. Right-click to add the user to the group. Log out and log back in for the changes to take effect.

The answer appears to be "no". I couldn't understand how everyone else at work had avoided this problem; but then learnt that they had asked for admin rights over their local machines and been granted them! I've only been here for two and a half years ... :|